Continuity equations and ODE flows with non-smooth velocity*

By Luigi Ambrosio, Gianluca Crippa. Proceedings of the Royal Society of Edinburgh. Section A Mathematics, 2014

171 citations recorded by the source corpus.

In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces (2015, background): “Uniqueness and in general the well-posedness of the Cauchy problem for advection equations like (1.4) are usually handled through the theory of renormalized solutions as introduced in [16] and improved in [1] (we refer to [2] and [15] for a very good introduction to the theory).”
Well-posedness of Multidimensional Diffusion Processes with Weakly Differentiable Coefficients (2015, background): “[3] for an account of the deterministic theory) is introduced, the main objects being Stochastic Lagrangian Flows, i.”